Finite Group Databasebeta

← #13054792

Subgroups — #13054792

Order 1344.

284 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621547C211yesmaximal, characteristic
2#10621425C211yesmaximal, characteristic, complemented
3C7×#944C211yesmaximal, characteristic, complemented
4C3×#6505C211yesmaximal, characteristic, complemented
5#10620515C211yesmaximal, characteristic, complemented
6#10620512C211yesmaximal, characteristic, complemented
7#10620497C211yesmaximal, characteristic, complemented
8#89867—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66728C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12C3×#1157C2211yescharacteristic
13#66683—22no—
14#66677—21no—
15D21⋊Q8—21no—
16#66670—21no—
17D7×Dic6—21no—
18#66635—21no—
19#66634—21no—
20#66627C2211yescharacteristic
21C7×Dic3⋊C4—22no—
22C4×C7×Dic3C2211yescharacteristic
23C3×#1144—21no—
24C3×#1143—21no—
25C3×#1136C2211yescharacteristic
26#6551—31no—
27C7×C4⋊Q8—31no—
28#6505—31no—
29#6505D311yescharacteristic, complemented
30#6410—31no—
31#6379—31no—
32#6379—31no—
33#5205—71nomaximal, Hall
34C3×C7×Q8—21no—
35C3×C7×Q8—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38#4268—21no—
39D84D411yescharacteristic
40C4×D21—21no—
41Dic42D411yescharacteristic
42C2×C7×Dic3—21no—
43C2×C7×Dic3D412yes—
44C7×Dic6—21no—
45C7×Dic6—21no—
46C3×C22⋊D7—21no—
47C3×D28D411yescharacteristic
48C3×C4×D7—21no—
49C3×Dic14D411yescharacteristic
50#4248—22no—
51#4246—22no—
52D21⋊C4—22no—
53Dic3×D7—22no—
54C3⋊(C7⋊C8)—21no—
55C3×C7⋊C8—21no—
56C2×C7×Q8—31no—
57C2×C7×Q8C2×D311yescharacteristic
58#1161—61no—
59#1161—61no—
60Q8×D7—61no—
61Q8×D7—61no—
62#1157—31no—
63#1157—62no—
64#1157C2×D311yescharacteristic
65C7×C4⋊C4—62no—
66C42×C7—31no—
67#1144—21no—
68#1144—61no—
69#1143—21no—
70#1143—61no—
71#1136—31no—
72#1136C2×D311yescharacteristic
73#1010—71no—
74C3×#137—71no—
75#944D711yescharacteristic, complemented
76#942—71no—
77#837—71no—
78#825—71no—
79#824—71no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D21—21no—
82C2×C3×D7—21no—
83C3×C4×C7—41no—
84C3×C4×C7C2×D411yescharacteristic
85C3×C4×C7C2×D411yescharacteristic
86Dic21—21no—
87C3×Dic7—21no—
88C7×Dic3—22no—
89C7×Dic3—22no—
90#456—211noSylow
91C7×Q8—21no—
92C7×Q8—21no—
93C7×Q8—61no—
94C7×Q8—61no—
95C2×C4×C7—21no—
96C2×C4×C7—32no—
97C2×C4×C7—61no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—21no—
100C22⋊D7—61no—
101D28—31no—
102D28—62no—
103D28#6711yescharacteristic
104C4×D7—21no—
105C4×D7—62no—
106C4×D7—62no—
107C4×D7—61no—
108Dic14—31no—
109Dic14—62no—
110Dic14#6711yescharacteristic
111C7⋊C8—21no—
112C7⋊C8—61no—
113C3×#41—71no—
114C2×C3×Q8C2×D711yescharacteristic
115D3×Q8—141no—
116#237—141no—
117#237—142no—
118#235—71no—
119#235—141no—
120C2×Dic6—141no—
121C2×Dic6C2×D711yescharacteristic
122C3×Q16—141no—
123C3×SD16—141no—
124C3×M16—71no—
125#216—141no—
126#215—141no—
127Dic3⋊C4—22no—
128C4×Dic3C2×D711yescharacteristic
129#208—71no—
130C2×C3×C7—21no—
131C2×C3×C7#12011yescharacteristic
132D21—41no—
133C3×D7—41no—
1342−1+4—211no—
135#137—71no—
136#137—211no—
137C4⋊Q8—31no—
138#104—211no—
139#104—211no—
140#101—211no—
141C22×C7D3×D411yescharacteristic
142C2×D7—21no—
143C2×D7—61no—
144C4×C7—41no—
145C4×C7—62no—
146C4×C7—62no—
147C4×C7D3×D411yescharacteristic
148C4×C7C2×#6711yescharacteristic
149Dic7—21no—
150Dic7—61no—
151C3×Q8—21no—
152C3×Q8—21no—
153C3×Q8—71no—
154C3×D4—71no—
155C3×D4—141no—
156C2×C3×C4—21no—
157C2×C3×C4—141no—
158C2×C3×C4C22×D711yescharacteristic
159#67—142no—
160#67—141no—
161C2×Dic3—21no—
162C2×Dic3—142no—
163C2×Dic3C22⋊D712yes—
164D12—71no—
165C4×D3—142no—
166C4×D3—141no—
167Dic6—21no—
168Dic6—21no—
169Dic6—71no—
170Dic6—142no—
171C3×C8—141no—
172C3⋊C8—141no—
173C3×C7#45611yescharacteristic, complemented, Hall
174#41—71no—
175#41—211no—
176#41—422no—
177#41—421no—
178#41—421no—
179C2×Q8—31no—
180C2×Q8—421no—
181C2×Q8—421no—
182C2×Q8D3×D711yescharacteristic
183Q16—141no—
184Q16—421no—
185SD16—141no—
186SD16—421no—
187M16—71no—
188M16—211no—
189C4⋊C4—62no—
190C42—31no—
191C2×C7—21no—
192C2×C7#93711yescharacteristic
193D7—41no—
194D7—121no—
195C22×C3—141no—
196C22×C3C2×C22⋊D711yescharacteristic
197C2×D3—141no—
198C3×C4—41no—
199C3×C4—141no—
200C3×C4D4×D711yescharacteristic
201C3×C4D4×D711yescharacteristic
202Dic3—22no—
203Dic3—22no—
204Dic3—141no—
205Q8—21no—
206Q8—21no—
207Q8—61no—
208Q8—61no—
209Q8—71no—
210Q8—211no—
211Q8—422no—
212D4—71no—
213D4—141no—
214D4—211no—
215D4—422no—
216D4—421no—
217C2×C4—21no—
218C2×C4—32no—
219C2×C4—61no—
220C2×C4—141no—
221C2×C4—422no—
222C2×C4—422no—
223C2×C4—421no—
224C2×C4C2×D3×D711yescharacteristic
225C8—141no—
226C8—421no—
227C7#520511yescharacteristic, complemented, Sylow
228C2×C3—21no—
229C2×C3—281no—
230C2×C3#650011yescharacteristic
231D3—281no—
232C22—141no—
233C22—421no—
234C22D3×C22⋊D711yescharacteristic
235C4—41no—
236C4—62no—
237C4—62no—
238C4—141no—
239C4—421no—
240C4D7×#6711yescharacteristic
241C4#6669411yescharacteristic
242C3#8986711yescharacteristic, complemented, Sylow
243C2—21no—
244C2—281no—
245C2—841no—
246C2#1062113911yescharacteristic
subgroupG/H
centerC2#10621139
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
prime radical#10621547C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#456
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #10621425C2semidirect
3. C7×#944C2semidirect
4. C3×#6505C2semidirect
5. #10620515C2semidirect
6. #10620512C2semidirect
7. #10620497C2semidirect
29. #6505D3semidirect
75. #944D7semidirect
173. C3×C7#456semidirect
227. C7#5205semidirect
242. C3#89867semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#106215472
#106214252
C7×#9442
C3×#65052
#106205152
#106205122
#106204972
#898673
#52057